USAPhO 2026, Първи кръг — Задача 9
2026 Exam · 1 т.
Указания
25 QUESTIONS - 75 MINUTES
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
- Use throughout, unless otherwise specified.
- You may write in this question booklet and the scratch paper provided by the proctor.
- This test has 25 multiple choice questions. Select the best response to each question, and use a No.2 pencil to completely fill the box corresponding to your choice. If you change an answer, completely erase the previous mark. Only use the boxes numbered 1 through 25 on the answer sheet.
- All questions are equally weighted, but are not necessarily equally difficult.
- You will receive one point for each correct answer, and zero points for each incorrect or blank answer. There is no additional penalty for incorrect answers.
- You may use a hand-held calculator. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones may not be used during the exam or while the exam papers are present. You may not use any external references, such as books or formula sheets.
- The question booklet, answer sheet and scratch paper will be collected at the end of this exam.
- To maintain exam security, do not communicate any information about the questions or their solutions until after February 13, 2026.
DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN
We acknowledge The US Physics Team coaches and other contributors to this year’s exams:
Tengiz Bibilashvili, Eddie Chen, Orhun Ciftcioglu, Kellan Colburn, Evan Erickson, Harry Kroft, Natalie LeBaron, Mai Qi, Elena Yudovina
Условие
A vertical cylindrical pipe with a sealed bottom contains no air and is tall enough that its top is open to outer space (no atmosphere). A heavy plate is mounted at the bottom of the cylinder and vibrates vertically. A small ball inside the pipe undergoes elastic collisions with the plate.
The plate vibrates in such a way that, during each collision, it is moving upward with a constant speed equal to , where is the orbital speed at zero altitude above the Earth. The plate moves downward sufficiently fast that the ball collides with the plate only while the plate is moving upward. The ball is initially dropped from rest from a height equal to the Earth’s radius above the plate.
After how many collisions with the plate will the ball escape the Earth’s gravitational field?
(A) 207
(B) 208
(C) 414
(D) 415
(E) 1000
Отговори
- B
Решение
At the first impact, the ball’s speed equals the escape speed from the Earth. In each subsequent elastic collision with the upward-moving plate, the ball’s speed increases by twice the speed of the plate. Therefore, after collisions the total increase in speed is .
Requiring the ball’s speed to reach the escape speed gives
so
After 207 collisions the ball still returns, but after the 208th collision it escapes the Earth’s gravitational field. Hence, the required number of collisions is .
Оригинал в Архива: 2026_FMA_exam.pdf · официални решения: 2026_FMA_solutions.pdf